Definition:Metrizable Topology
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Definition
Let $\struct {S, d}$ be a metric space.
Let $\struct {S, \tau}$ be the topological space induced by $d$.
Then for any topological space which is homeomorphic to such a $\struct {S, \tau}$, it and its topology are defined as metrizable.
Also see
- Indiscrete Topology is not Metrizable: thus, not all topological spaces are metrizable
- Results about metrizable topologies can be found here.
Linguistic Note
The UK English spelling of metrizable is metrisable, but it is rarely found.
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Metrizability